Here is one way to construct such a table. Start by satisfying only the second part of the condition, that the products by columns are equal. This holds for the first table $T$ below. It has dimensions $4 \times 4$ and all column products equal $120$. The second table has an additional column in which the row sums of $T$ are shown.

$$T = \begin{array}{c|c|c|c} 4 & 1 & 2 & 8 \\ 3 & 2 & 5 & 1 \\ 2 & 10 & 3 & 5 \\ 5 & 6 & 4 & 3 \end{array} \rightarrow \begin{array}{c|c|c|c|c} 528 & 132 & 264 & 1056 & 1980 \\ 540 & 360 & 900 & 180 & 1980 \\ 198 & 990 & 297 & 495 & 1980 \\ 550 & 660 & 440 & 330 & 1980 \end{array}$$

Note that the equality of the column products is preserved if a row of $T$ is multiplied by any number. We use this observation to equalize the different row sums by choosing an appropriate integer for each row. To this end consider the least common multiple $1980$ of the row sums, $15$, $11$, $20$, $18$.

Multiply rows 1, 2, 3, 4 respectively by $\frac{1980}{15} = 132$, $\frac{1980}{11} = 180$, $\frac{1980}{20} = 99$, $\frac{1980}{18} = 110$. The new table has row sums equal to $1980$ and equal column products (their common value is the large number $31049568000$). It remains to observe that all new numbers are different.