It's impossible to say for sure unless you describe how the similarity scores were computed.
In general, for the usual kind of similarity scoring this is not possible: information has been lost in the transformation from individual features to aggregate statistics. The best you can hope to do is to arrive at a set of features that are consistent with the similarity scores.
I think that is what you are talking about when you say "similar to" the original. That problem is pretty interesting. Suppose similarity was computed as the dot-product of two feature vectors (ie the count of features for a pair of objects that both have value = 1/true). This is not the only choice: it is consistent with value of 0 (false) meaning no information. But it may generalize to other similarity measures.
In such a case, the problem is really a linear programming problem: a naive approach is to exhaustively search the space of possible objects - not randomly, but guided by the constraints. For example, suppose SIM(A,B) := similarity of object A and object B. Define an order on these vectors.
If SIM(A,B) = N, then choose A=B minimal (like (1,....,1 (N times), 0, .... 0 (1000-N times)), and then choose the minimum C s.t. (A,C), (B,C) have the given values. Once you find an inconsistency, backtrack, and increment.
This will find a consistent answer, although the complexity is very high (but probably better than monte carlo).
Finding a better algorithm is an interesting problem, but more than this I can't say in a SO post - that's probably a topic for a CS thesis!