The Beastmaster Ranger's Ranger's Companion feature states:

[...] If the beast dies, you can obtain another one by spending 8 hours magically bonding with another beast that isn't hostile to you, either the same type of beast as before or a different one...

This is the only way mentioned of acquiring a novo companion, so unfortunately if that is not an option, you will not have a way of getting a new companion. That said, you do have a few options available:

Finding a new companion

Existe a locate animals or plants soletrar:

Describe or name a specific kind of beast or plant. Concentrating on the voice of nature in your surroundings, you learn the direction and distance to the closest creature or plant of that kind within 5 miles, if any are present.

Preventing the companion from dying:

Suggest to your GM that they allow your animal companions to make death saving throws. This would also give you more time to react once your companion begins to "die", potentially saving them. And it expressly mentioned in the following section:

Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws. Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.

Você também pode usar o ala da morte soletrar:

You touch a creature and grant it a measure of protection from death.

The first time the target would drop to 0 hit points as a result of taking damage, the target instead drops to 1 hit point, and the spell ends.

If the spell is still in effect when the target is subjected to an effect that would kill it instantaneously without dealing damage, that effect is instead negated against the target, and the spell ends.

This spell lasts for eight hours, and so a Cleric/Paladin in your party may cast it for you, otherwise perhaps there is a spell-selling service in your world.

Pretend it is magical

This isn't RAW, but I have been in a similar situation and the DM worked a situation where the animal was 'blessed' (In our case by a powerful dragon) and bonded to the ranger in such a way that the spirit could be called in a very ache familiar-ish manner.

This meant that the ranger was never more than an hour away from having a companion.

Note: It is important to take into account the role play implications of this. A vulnerable companion can lead the ranger into making personal sacrifice for their animal, but an effectively indestructible companion would end up being played the other way round. This definitely isn't a solution for all tables.

Is there realistically anything I could have done or do Beastmaster rangers have to accept that if their beast companion dies they will be sitting on the sidelines until they can replace it?

RAW, it's the latter. Risking your resources against challenges is a big part of the game.

Beastmasters get some special advantages from their class features, as do any characters with class levels. Whether or not those features are balanced with one another across classes (by which I mean, is a Wizard as "good" as a Beastmaster) is a more nebulous question, but each class still specializes in certain ways and can be in trouble if something precludes their specialized capabilities.

For a Wizard, it's all about spells. If a Wizard PC chooses a not-very-useful set of spells before a dungeon dive that doesn't allow a long rest, they will be less effective than they might have been and have a harder time than they might have had. If the Wizard burns their spell slots early, they'll have a lot less utility later on. If they find themselves in an area filled with antimagic fields, or encountering lots of enemies with magic resistances, then they'll be much less valuable in their typical role.

For a Beastmaster Ranger, the specialization is in the beasts that they master. Your animal companion dying, while not necessariamente preventable, is a situation akin to a pure Wizard being silenced, in an antimagic field, or in some other way having been prevented from using their spells. Especially if you're specialized around the animal companion, keeping that companion alive is as important for the Beastmaster as spell management and ability to cast spells are to Wizards. This dungeon may have been especially difficult for your character, since no suitable replacements were available.

In any of these cases the player's choices, and their circumstances and luck, impact which of their resources they hazard at a particular time. It may be necessary to risk an animal companion (or a PC companion!) to overcome an obstacle, and then you have that much less to bring to bear on future challenges. That sort of resource allocation is a very important element of the game, and not every risk pays off.

I say all of this not to put down the Beastmaster class specialization, but to highlight that the animal companion is a high-value resource which may not always be available. In the same way that a Wizard will use spell slots to advance in the dungeon and then not have them available afterwards, a Beastmaster's companion will help the party advance but might be "consumed" in doing so. It's easily fixed during downtime, but in an adventure featuring time pressure the inability to get a new companion might be similarly vexing as a Wizard's inability to take a long rest and restore spell slots-- whittling away at the party's resources is part of the adventure's challenge.

What can you do about the Beastmaster's vulnerability to losing their animal companion?

Having a list of options to resurrect the companion is valuable, though doing it mid-dungeon crawl may require assistance and investment from others in your party.

Being slightly less specialized around the companion may also be an option, depending on your tastes. Going all-in on the companion is an all-your-eggs-in-one-basket proposition, and the risk of that is that they all break at once and you're down to crossbow fire.

Mais importante ainda, fale com o seu mestre. The key angle is that your character is based around the animal companion, and presumably a good portion of your fun is related to that as well. Because the companion is a special class resource that can't be restored as easily as others (like by simply taking a long rest), your character may be particularly brittle relative to the other PCs. If that's impacting your fun, as seems to be the case, your DM may be willing to work with out to remove some of that brittleness.

I disagree with your DM's call that only PCs can make death saves. This is explicitly contrary to the published rules, which make no such restriction, but even if this were untrue (or your DM simply handles their table that way) a special accommodation could easily be made for the animal companion. Other options exist to make your character build a bit more resilient, such as UA's Revised Ranger allowing for a particular fallen companion to be restored over a long rest.

What can you do, generally, when your specialization is ineffective or unavailable?

There's no question that being deprived of your character's specializations takes some of the excitement out of the game-- those are what make your character shine! But even if your character is not able to do what they do best, they can still do a lot to help out:

• Healing is a grande negócio. Many combat situations expect the party to replenish HP, and even if it isn't your favorite thing there's no cause to feel bad about "just" healing.
• Helping. It's situational, but using the Ajuda action to assist party members can also be valuable. Advantage is a meaningful benefit.
• Drawing fire/Engaging enemies. If your character has good AC, and/or can take a hit, engaging an enemy can be really useful. It potentially limits enemy mobility, and helps spread out enemy damage output rather than concentrating it on the rest of the party. I've had players do nothing but Dodge for rounds at a time, soaking up enemy attack actions without actually taking damage.
• Attacking. Damage adds up, and a single crossbow bolt can make the difference between a successful fight and a catastrophe.
• Grappling. If your character has the attributes for it, grappling an enemy can take them out of the fight for a time.

Resurrect it

In the immediate moment, an animal companion is a perfectly valid target for the spell reviver, which is available to clerics (and Celestial-patron warlocks) from 5th level and paladins from 9th. Revivificar is so generally useful in an emergency that once the party has access to it, it would be quite remiss for nobody to have it prepared, and to try and leave at least one 3rd+ spell slot available to cast it. Fights are usually concluded inside of a minute so it should be a rare case that the caster cannot get to a fallen ally/companion in time to revive them.

This is a relatively expensive option as it costs 300gp of diamonds to reviver a creature. If you miss out on the minute window and have to go up to ressuscitar morto instead, it requires a 500gp diamond and your companion will be quite nerfed until they've had a few rests to recover. In the event that the party is of the particularly stingy kind who might balk at spending that amount of money on resurrecting an animal, your ranger might want to consider getting ahold of the necessary jewellery with their own money so it can be used for material components should the situation arise.

Is there realistically anything I could have done or do Beastmaster rangers have to accept that if their beast companion dies they will be sitting on the sidelines until they can replace it?

DM flatly refused death saves on the animal companion stating it was not a PC.

I'm afraid the answer is pretty simple. Don't play a Beastmaster ranger when the DM's choice of rules will predictably make your companion regularly die.

Being reduced to 0 HP in a 5e fight is pretty typical against challenging foes, and your beast companion isn't ridiculously tough like a barbarian or moon druid.

So your beast companion is going to regularly be reduced to 0 HP if your fights are ever tough. If you aren't playing in environments where you can easily get a new companion, and death saving throws aren't allowed, you will regularly be "useless" in the sense of "not have an animal companion".

And it is a matter of DM's choice of rules, not "playing by RAW" or whatever:

The 5e rules state that PCs get saving throws, monster's generally don't, and Mighty villains and special nonplayer characters are common exceptions; and get death saves. A beast companion is about a special as a nonplayer character gets.

So, the DM is free to state "beast companions die at 0 HP", but that just means you should avoid playing Beastmaster Rangers in that DM's world if you want to tell the story of a competent adventurer.

Find a style of PC your DM supports as being competent, and play that. I'd avoid being creative in your second choice of PC with that DM, as you don't want to run into having yet another PC be proven useless due to rules interpretation. Instead, just pick a type of PC that other characters have already chosen to be acceptable to the DM.

Tell the rest of the party that you necessidade to leave the dungeon to get a new animal companion, and do not compromise.

Just like a Wizard who has run out of spells, you need to retreat from the dungeon and take a long rest to regain the functionality of your class - and Ranger is already widely regarded as the weakest class in the game to begin with. If the rest of the party doesn't comply, just have your character turn around and leave without them. If the party doesn't follow along you or otherwise take a long rest so that you can regain the party, just tell them that the alternative is having your character kill himself so that you can roll up a new character, and thereby delay the game while you do that, or for you to get up and go home while your character is off by himself and the rest of the party ignores him.

This isn't the best option, or the first one you should try, but if they're not going to listen to reason, then the only option you have is to twist their arms. If they don't comply even, then, just get up, tell them that you hope they'll enjoy the rest of the night, and leave the game and go home. No DnD is better than bad DnD.