For small $m$, I've computed the cardinality $s(m)$ of the largest subset of $\{0,1\}^m$ of Hamming diameter $< m/2$, as follows:

$$\matrix{m & s(m)\cr
1 & 1\cr
2 & 1\cr
3 & 2\cr
4 & 2\cr
5 & 6\cr
6 & 7\cr
7 & 14\cr
8 & 16\cr}$$
The sequence does not appear to be in the OEIS. Clearly this is much too small
for asymptotics (and with the methods I'm using it's going to be hard to get results for much larger $m$), but some of the sets returned seem to have a rather interesting structure. For example, for $m=8$ here is a set of size $16$ that has diameter $3$:

$$ \matrix{00000001\cr
00000010\cr
00000011\cr
00000101\cr
00000110\cr
00000111\cr
00001011\cr
00001111\cr
00010011\cr
00010111\cr
00100011\cr
00100111\cr
01000011\cr
01000111\cr
10000011\cr
10000111\cr
}$$

EDIT:
One lower bound of $s(m)$ is the cardinality of a ball of radius $\lfloor (m-1)/4 \rfloor$. This is $2^{m} \mathbb P(S_m \ge m - \lfloor (m-1)/4 \rfloor)$ where $S_m$ is the number of heads in $m$ independent fair coin flips.
By Cramér's theorem from Large Deviations theory, as $m \to \infty$ this is roughly
$ 2^m \exp(-m I(3/4)) = 2^{-km}$ where $k = 2 - (3/4) \log_2(3) \approx 0.811278$. Thus if $m$ is sufficiently large, a ball of radius $\lfloor (m-1)/4 \rfloor$ provides a counterexample.

Numerically, $377$ seems
to be the least $m$ for which this works. That is,
a ball of radius $94$ in $\{0,1\}^{377}$ has diameter $188 < 377/2$ and cardinality approximately $6.315655200 \times 10^{90} $, which is
more than $2^{0.8 \times 377} \approx 6.175138852 \times 10^{90}$.

Just to show this isn't an artifact of roundoff error, the precise cardinality in this case is
$$ \sum_{i=0}^{94} {377 \choose i} = 6315655199547126133494801496762823097854137171340937564314538960656350969227380734836894104$$

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