Yes, it is possible

It is actually possible to obtain a permanent skeleton companion. That being said, it is a slightly convoluted process that is not without certain shortcomings. This method involves using two spells in conjunction:

Finger of Death

A humanoid killed by this spell rises at the start of your next turn as a zombie that is permanently under your command, following your verbal orders to the best of its ability.

True Polymorph

Choose one creature or nonmagical object that you can see within range. You transform that creature into a different creature ... If you concentrate on this spell for the full duration, the transformation becomes permanent. ... If you turn a creature into another kind of creature, the new form can be any kind you choose whose challenge rating is equal to or less than the target's. The target's game statistics, including mental ability scores, are replaced by the statistics of the new form. It retains its alignment and personality. ...

By now, you probably understand where I'm going with this. You would first start by using Finger of Death to create a normal zombie servant under your control. Next, you would use the True Polymorph spell to transform said zombie into a skeleton (which you can do because they are both challenge rating 1/4). Then you simply wait out the spell's duration, which makes the transformation permanent, and, since a transformed creature "retains its alignment and personality" you now have one permanent skeleton companion at your disposal.

In fact, using this method, you could create a permanent companion out of any creature of 1/4 challenge rating you may desire.

In Summary

Finger of Death + True Polymorph = permanent skeleton companion

Unarmed attacks can crit, but do not do double damage

• Can an unarmed strike crit? - Yes

  If you roll a 20 on an unarmed strike, it is considered a critical hit.

  This means that it will automatically hit regardless of the opponent's AC and has the potential to cause 2 failed death saving throws from a creature at 0 HP.

  However, a critical hit does not get to increase its damage from a critical hit.

• Do unarmed strikes get double damage on a critical hit? - No

  Critical hits only double damage that you roll. Unarmed strikes do not have any damage rolled. So, RAW, critical hits do not increase the damage of unarmed strikes. See this Q&A for further discussion of this.

  Jeremy Crawford agrees and has (unofficially) clarified:

  An unarmed strike deals 1 + Str. modifier damage, even on a critical hit.

  This ruling, besides being in line with the literal reading of the RAW, is likely made to keep the rules of what damage to multiply from a critical (only rolled damage) consistent. If you have to start carving out exceptions for every static damage effect that doesn't roll damage but you think might kind of be like a roll, applying the critical hit rule could get complicated and cumbersome. Thus the simple, easy-to-apply ruling we get here.¹

Does the described attack cause failed Death Saving Throws? - No

• It is possible for an attack to do 0 damage

  Jeremy Crawford has (unofficially) clarified:

  There is not a damage minimum of 1, so it is possible to deal 0 damage with an attack.

  So you are indeed calculated the damage for that attack correctly.

• 0 damage does not count as damage

  Jeremy Crawford has (unofficially) clarified:

  Taking 0 damage is the same as taking no damage.

• No damage means no failed death saving throws

  If an Unconscious character takes damage while at 0 HP, they automatically fail one death saving throw, or 2 death saves if the damage is from a critical hit.

  If a creature takes damage while at 0 hp, then they will fail one saving throw (or two for a crit). If no damage is taken then no death saving throws are failed. The critical does not change this because damage is a prerequisite in order for any death saving throws to be failed in this way.

So the character in this case does not fail any death saving throws as a result of the attack

^{1 - Note that I've played in and run games where I've allowed unarmed strikes to double damage as a house rule, and it seems to have had only positive effects (if any, since it came up extremely rarely). But it was always made very clear that this exception was being made for only this one thing. If you did so at your table, I'd guess you wouldn't have any issues either. The only thing to watch out for is confusion I'd say, because the rule that only rolled damage is rerolled is a handy one, and breaking that for unarmed strikes might undermine that. I've only done this house rule with experienced players, so keep that in mind.}