Is it possible to encrypt quantum states under reasonable assumptions? Is it possible to encrypt a quantum state, such that a $BQP$ attacker who does not know the secret key cannot obtain any information about the original state, but a $BQP$ decryptor with the key can recover the original state?

Assume (classical) one-way functions that are hard for quantum attackers to break exist (as is believed to be the case).
One can encrypt an n-qubit state using a 2n-bit classical secret key. The idea is to use the key to select a random Pauli operator, and apply that operator to the secret as an encryption. (The inverse operator is applied to decrypt.)

The resulting scheme is perfectly secure -- if the key is selected uniformly at random, then even an attacker who know a state entangled with the plaintext cannot distinguish the true ciphertext from an independent random state. 

This observation was first made in
A Ambainis, M Mosca, A Tapp, R de Wolf. "Private quantum channels", FOCS 2000.

They also showed that 2n bits of key are necessary for encryption of entangled quantum states. 

If you have a classical PRG secure against quantum distinguishers, then you can get away with a much shorter key (as short as seed length required by the generator to generate 2 n output bits).