Is it possible to replicate the ENIAC using logic gates
The ENIAC used logic gates, they were just made up of vacuum tubes.  The ENIAC had thousands of them. 6,550 out of the 18,800 tubes in the computer (circa 1946) were dual-triode 6SN7GTs. Each tube was used as a flip-flop and represented one bit, one side the 1, and the other side the 0.

The ENIAC used ten-position ring counters to store digits in ten's complement representation. Each digit used 28 vacuum tubes, 10 of which were the dual triodes making up the flip-flops of the ring counter (the ten tubes shown on the right in the picture below).



Arithmetic was performed by counting pulses with the ring counters and generating a carry if the counter wrapped around. ENIAC had twenty ten-digit signed accumulators.



So you could start out by building one of the accumulators.  You could cheat and just use a ring-counter IC, such as a CD4017; but to be more faithful, you should build one out of ten flip-flops plus additional logic as needed. To really capture the original design, you could build one out of transistors.

Constants were entered using punched cards; programming was done using a combination of rotary switches and jumper cables.



This document has a pretty good description of how it worked, including timing diagram as shown below.  Note a '1' was represented by -290v, and a '0' by -345v.



This article contains many of the electronic circuits used in the ENIAC, and here is the user's manual.