Minimum storage regenerating (MSR) codes are MDS codes which allow for recovery of any single erased symbol with optimal repair bandwidth, based on the smallest possible fraction of the contents downloaded from each of the other symbols. Recently, certain Reed-Solomon codes were constructed which are MSR. However, the sub-packetization of these codes is exponentially large, growing like $n^{Ω(n)}$ in the constant-rate regime. In this work, we study the relaxed notion of $ε$-MSR codes, which incur a factor of $(1+ε)$ higher than the optimal repair bandwidth, in the context of Reed-Solomon codes. We give constructions of constant-rate $ε$-MSR Reed-Solomon codes with polynomial sub-packetization of $n^{O(1/ε)}$ and thereby giving an explicit tradeoff between the repair bandwidth and sub-packetization.