Would a paired t-test be appropriate if the variables had a range between 0 and 1? Suppose I have paired data $(X_1, \ldots, X_n)$ and $(Y_1, \ldots, Y_n)$ and I would like to compare if the means $\bar{X} = \frac{1}{n}\sum_{i=1}^nX_i$ and $\bar{Y} = \frac{1}{n}\sum_{i=1}^nY_i$ are significantly different.
The range of each variable is between 0 and 1: $X_i \in (0,1)$ and $Y_i \in (0,1)$.
In this case, would a paired t-test be appropriate due to the fact they range from 0 to 1?

Yes, provided that $n$ is sufficiently large for the approximate (asymptotic) normality of $\bar{X}$ and $\bar{Y}$ to hold.
It's difficult to say exactly what value of $n$ is appropriate here, but if you make, say, a confidence interval (say, 95% or 99%) for the two population means, using the standard set up:
$$\bar{u} \pm t \sqrt{ \frac{s_u^2}{n} }$$
where $\bar{u}$ is the mean of the sample, $s_u^2$ is the sample standard deviation, $n$ is the sample size and $t$ is your relevant critical value, and neither interval spills outside of $(0,1)$, I'd imagine you would be fine
