No; it can't enter. It is a magical effect
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.
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.
Leomund's Hut blocks magical effects.
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.