As can be read in the first few chapters of Harry Potter and the Philosopher’s Stone, Harry is deposited at the doorstep of his aunt and uncle, Petunia and Vernon Dursley. Petunia Dursley and Lily Evans were sisters and therefore the Dursleys constitute the only living relatives of Harry Potter.
The following ten years are omitted from the story in exact terms, although the books divulge some details about Harry’s stay with the Dursleys. Problems include being clothed with hand-me-downs from Dudley, being denied food on several occasions, the occasional beating, allowing Dudley and his social circle to harass Harry, excluding Harry from swimming lessons, and letting Harry sleep in a cupboard under the stairs.
Any social worker hearing of this situation would reek the child abuse instantly and would descend onto the situation with a fervour. As it did not happen, I am afraid I have to launch a few of my pet theories with regards to the situation: either social services in the Potterverse are ignorant and stupid, or even non-existent, or they are kept from taking action. The first few options are horrifying enough, but the second idea, that the social services are kept from taking action, deserves more attention, as that branches out into several possible causes as well.
There are two persons who consider it of great importance this situation remains under the radar: Albus Percival Wulfric Brian Dumbledore and Vernon Dursley.
Vernon Dursley is especially suspect, as he is the one causing the abuse and has the financial power to keep the social services from finding out. If he is responsible for that, it also solves other problems, for example why the Dursleys haven’t moved over the years despite having considerable wealth, as Vernon is the CEO of Grunnings. At the start of the story Vernon and Petunia have recently married and it may be expected that Vernon occupied that position for only a short time. Ten years should be enough to save up money for a bigger and better house. It could be that a lot of his earned money is siphoned off to keep the social services away from his house or from taking action.
Another far grimmer possibility is that Dumbledore uses this to keep Harry with his relative in order to keep the blood wards intact at all costs by not having him removed from there.
Which ever of these assumptions, or any other, is most correct, depends on the corners of the internet which you, dear reader, frequent.