It isn't a small interval, is it?

The real numbers which have more than one decimal representation are the ones ending in an infinite string of $9$ 's, e.g.

$0.\overline{9} = 0.99999.... = 1.00000.... = 1 \\ 0.32\overline{9} = 0.32499999.... = 0.32500000.... = \dfrac{13}{40} \\ 7.\overline{9} = 7.19999.... = 7.200000.... = \dfrac{36}{5}$

Every one of these numbers can be thought of as a terminating decimal with an infinite string of $9$ 's "attached"; therefore there is a one-to-one correspondence between the set of real numbers with more than one decimal representation and the set of terminating decimals. The set of terminating decimals in the interval $(0, 1)$ is clearly infinite, and as a subset of the rational numbers $\mathbb{Q}$ must also be countable; thus the set of real numbers in $(0, 1)$ with more than one decimal representation must also be infinite and countable.