We need to form a number:

• greater than $1000$
• less than $4000$
• using digits $0,1,2,3,4$
• repetition allowed.

Since the number is between $1000$ and $4000$, it must be a 4-digit number.

The thousands digit determines the range:

• It cannot be $0$ (otherwise not a 4-digit number).
• It cannot be $4$, because then the number would be at least $4000$, which is not allowed.
• So the thousands digit can only be $1,2,3$.

Hence, number of choices for the first digit = $3$.

For each of the remaining three places (hundreds, tens, units):

• any of the digits $0,1,2,3,4$ can be used,
• repetition is allowed.

So each of these places has $5$ choices.

Total number of such numbers: $3 \times 5 \times 5 \times 5 = 3 \times 125 = 375.$

Check the boundary condition:

• $1000$ itself is included in this count, but the question says greater than $1000$, so exclude $1000$.
• $4000$ is not included anyway, since leading digit $4$ was not allowed.

Therefore required count is: $375 - 1 = 374.$

Hence the correct option is: $\boxed{374}$ which is Option C.