Not astronomical gains. Ignoring the technical challenges of a biological system like DNA, base 4 can encode the same data in exactly half as many characters, so capacity would double.
X digits of binary vs quaternary can represent 2^x vs 4^x possibilities.
Even at 10 digits, that is 2^10 = 1024 vs 4^10 = 1048576.
with base2 we get 2^5 = 32 possible combinations
with base4 we get 4^5 = 1024 possible combinations
2*(2^x) != (4^x)
Take, for example, 32bit integers vs 64bit integers (unsigned for simplicity). Two 32 bit integers can represent exactly the same number of combinations that a single 64 bit number can. Sure, there's an exponential number more combinations in a 64bit integer than a 32 bit, but the number of combinations is not how the storage capacity is measured.