So it's quite obvious that a countably infinite collection of independent Bernoulli random variables exists but as always uncountability complicates the matter.
For similar questions about uncountable collections of non-constant random variables on $[0,1]$ the answer is no, see here. But I could not find any resources talking about uncountable collections of Bernoulli random variables.
It is not clear whether you are interested only in random variables on $(0,1)$ or any probability space will do. Given any collection of distribution functions $\{F_i\}_{i \in I}$ we can construct independent random variables $\{X_i\}_{i \in I}$ such that $X_i$ has distribution $F_i$ for each $i$. Kolomogorov's Consistency Theorem covers this and even more general constructions.