While it is true that there is no such spell as living unseen servant described the the published materials, that does not mean it is flat out not possible. Why, the world full of possible things that were not possible at some point.
As always, it depends on the DM. If the DM would allow such a thing, it might be possible. If you are asking as a player, talk to your DM. If you're asking as a DM, you have to ask yourself if you would want such a thing in your campaign.
After all, every spell was presumably invented by someone. You could even argue that either a) the world contains infinite examples beyond the power of even the most powerful player character, which personally I think is pretty lame or b) somehow, some way, with enough effort, what the PCs see, they could aspire to do.
Doesn't mean it's easy.
In Xanathar's Guide to Everything, a downtime activity of research is described (p. 132-133). A character could attempt to research an answer, or an answer, or even just information or a hint.
The time and cost of research is measured in weeks and hundreds of gold for even a few pieces of lore, and typically requires access to a library or a sage.
Xanathar's also describes crafting a magic item (p. 128-129). While apparently intended for creating items described in the published materials, it is not such a stretch that a DM might allow a character to craft a magic item that is different from the ones described in the books.
The description for crafting magic items includes:
Creating a magic item requires more than just time, effort, and materials. It is a long-term process that involves one or more adventures to track down rare materials and the lore needed to create the item.
It is not such a stretch that a character could research a new spell.
Such an endeavor might be similar, or a combination of, lore research or crafting a magic item. Of course, the DM could choose to make such an effort easier or harder than that. A wizard can create an unseen servant, create a homunculus, create undead, capture the souls of others. Surely somewhere in there creating a living unseen servant is not completely out of the same league.
Making the effect permanent, that does seem like it's much harder. Maybe that will take more research.
And of course, there is wish. A wizard could attempt to wish for a living unseen servant, or wish for a servant to become unseen, or wish an existing living unseen servant to acknowledge the wizard as master, or many other permutations.
The are other possibilities, which, are admittedly, a bit of a stretch, but maybe not completely. Clerics can call for divine intervention. Sorcerers have powerful patrons. In either case, perhaps a character could gain a permanent living unseen servant as part of a (no doubt bad) bargain of some sort.
As another answer said, this answer may be a bit unsatisfying, and is definitely pushing at the boundaries of RAW, but I don't think it's out of the bounds of RAI. The game is intended to be extensible and customizable, and this is definitely one of those cases.