Yes, nobody's in the house is correct.
In English, the meaning of nobody's depends on the context.
The subject-verb nobody is can be contracted to nobody's, as in nobody's perfect and nobody's in the house.
The contraction nobody's is less formal and is usually used in conversational English. The uncontracted nobody is is more formal and is usually used in writing.
The possessive indefinite pronoun nobody's expresses possession, as in that is nobody's business (that is the business of nobody) and nobody's cat is in the house (the cat of nobody is in the house).
- The possessive pronoun nobody's is used in conversational English and in writing.