Are there Similar Distance Binary Error Correcting Codes?
Every $\epsilon$-biased set gives a code whose minimal relative distance is $0.5 - \epsilon$ and maximal relative distance is $0.5 + \epsilon$.

To see it, write the elements of the set as the rows of a matrix, and then define the code to be the span of the columns of the matrix. The $\epsilon$-biased property of the set is equivalent to saying that the relative distance between codewords is always between $0.5 - \epsilon$ and $0.5 + \epsilon$.

One particular construction of such sets will give you a code whose dimension is linear in its block length. It is mentioned here:
http://www.wisdom.weizmann.ac.il/~benaroya/SmallBiasNew.pdf

Basically, the idea is to take AG codes of constant rate and relative distance close to $1$, and concatenating them with the Hadamard code.