Is it legal to hire someone as a personal assistant and pay them via living space?
It's not illegal but it is complicated. The problem you would run into is that you will owe tax and employer withholding/social security. There are some jobs that require you to live in such as a super in a building. In that situation there would be no taxable benefit attached to the accommodation. But if you were to exchange living space for office work, the value of the benefit is considered income to your employee and the imputed rent taxable to you. There is also the difficulty that the person is likely to take increasing advantage of you over time so the work will degrade. Simpler and better to unbundle this. Pay the person for the office work they do by the hour, day or week which you can log. Charge a market rent with a written lease and rules (month to month is ok).
It is generally legal for housing to be considered PART of the compensation. That said, I seriously doubt the fair market value of housing would come anywhere close to what you would have to pay a personal assistant.
Providing they are free to walk away at any time, its good. Mind you they would need to get paid something as we all have needs, like sending a birthday card or buying a phone or a hair cut.
Pascal the Gambler
So long as the value of said space works out to minimum wage, absolutely. They'd essentially be paid the value of the rent for such a space.