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The spell Unseen Servant creates a "Medium force" that can move around and perform simple tasks.
Meanwhile, the spell Leomund's Tiny Hut blocks creatures & objects from passing through it, but not "forces".
Can an Unseen Servant move into a Tiny Hut?
The key to this ruling is in what spells do, or dont' do.
What is a spell?
A spell is a discrete magical effect (Basic Rules, p. 82)
If the spell created or summoned a creature, the way that Animate Objects creates creatures, it would say so. It does not. The result is a magical effect that takes the shape of "a Medium force." Unlike a creature this Medium Force (1) can't attack (for example), and (2) doesn't have a complete stat block.
Unseen Servant
This spell creates an Invisible, mindless, shapeless, Medium force that performs simple tasks at your command until the spell ends. It has AC 10, 1 hit point, and a Strength of 2, and it can’t attack. If it drops to 0 hit points, the spell ends.
Note that at 0 HP it doesn't die, but the spell (the magical effect) ends.
If these magical effects originate outside of the Hut, they can't get in.
Spells and other magical effects can't extend through the dome or be cast through it.
The Unseen Servant remains outside of the Hut. Even if it were a creature, it would not be able to enter if it began outside of the hut.
If there were an exception to this, the spell text for the Hut, or for Unseen Seervant, would note that execption.
There are some caveats for your GM to consider. This is an edge case of the rules where the language is ambiguous. RAW and RAI, the text of Leomund's Tiny Hut only suggests that magical effects of any sort cannot enter the Hut.
Creatures and objects within the dome when you cast this spell can move through it freely. All other creatures and objects are barred from passing through it.
All is good with this line because Unseen Servant is a "Medium force".
Spells and other magical effects can't extend through the dome or be cast through it.
This is where the room for interpretation begins to arise and each GM will come to their own conclusions.
Many spells which have permanent effects have a duration of Instantaneous; with a duration of 1 hour, the medium force may be interpreted as an ongoing and probably magical effect of the spell Unseen Servant. Because of this, it doesn't seem like Unseen Servant was intended to be able to enter the dome or be cast on the opposite side of the caster. But this is where the GM will probably need to make a decision regarding the meaning of "extend".
A "Medium force" is produced by Unseen Servant, and Leomund's Tiny Hut produces a "dome of force". Some GMs may safely use this to rule that the servant cannot pass the dome. However, if the GM permits Unseen Servant to move through solid material it may enter through the ground below and bypass the dome.
If you're a player, I recommend asking your GM to find out how they would rule this before you attempt it at the table (and so that you don't spend precious known spells on an effect that isn't part of the rules & rulings).
