The answer to this is "yes in theory, probably not in practice".
A Fully Homomorphic Encryption (FHE) scheme is a traditional encryption scheme with an additional "evaluation" algorithm.
This means that, given a ciphertext $\mathsf{Enc}_{pk}(m)$, you can (for any circuit $C$) compute:
$$\mathsf{Eval}_{pk}(C,\mathsf{Enc}_{pk}(m))$$
to get an encryption of $C(m)$. Note that $\mathsf{Eval}$ is a publicly computable function, but you are operating on encrypted data the whole time, e.g. this notion seems to be (precisely) what you want.
The issue with FHE is mainly of efficiency. The programs evaluated with FHE are generally represented as circuits (rather than arbitrary turing machines).
This means that the control flow of the program is fixed, so for an if statement you have to evaluate both branches of the computation.
For certain standard constructs (say HashMaps), this limitation means that I don't think anyone really knows how to implement them with FHE (without taking a massive efficiency hit). Depending on the precise program you want to evaluate this may be prohibitive.
There are definitely applications where FHE is currently feasible (mostly in computing statistics of encrypted data), but to evaluate the feasibility of what you want we would need to know a better description of precisely what you want to do, and for general-purpose computing the answer tends to be "it is infeasible" currently.
