We need the number of functions $f:\{1,2,\ldots,100\}\to\{0,1\}$ such that exactly one of the positive integers less than or equal to $98$ is assigned the value $1$.

The elements "less than or equal to $98$" in the domain are: $\{1,2,\ldots,98\}$ There are $98$ such elements.

Condition: among these $98$ inputs, exactly one must satisfy $f(x)=1.$ So we choose which one of the $98$ elements gets mapped to $1$.

Number of ways: $\binom{98}{1}=98.$

For the remaining elements of the domain, namely $99$ and $100$, there is no restriction. Each of these can independently be assigned either $0$ or $1$.

So for $99$ and $100$, the number of possibilities is: $2\times 2 = 2^2 = 4.$

Hence total number of functions is: $98\cdot 4 = 392.$

Therefore, the required number of functions is $\boxed{392}.$