Take the bag with the most apples, and set it aside. Let's say it had $A$ apples. Do the same with the one with the most beets, and say it had $B$ beets. We ignore these until the very end.
Queue up the remaining $98$ bags in order of number of apples. We split these $98$ bags into a Left and Right pile, using the following algorithm:
While the queue is nonempty, remove two bags at the end with fewer apples.
Of these two bags, call the bag with more beets X, and the other Y.
Throw X into either the Left or Right pile, whichever has FEWER beets.
Throw Y into the other pile.
I claim that, when this is done, that the "apple discrepancy" between the Left and Right piles is at most $A$, and the "beet discrepancy" is at most $B$.
Let's say the apple sizes were $a_1\le a_2\le \dots\le a_{98}$. In the worst case, the apple discrepancy is $$(a_{98}+a_{96}+\dots+a_2)-(a_{97}+\dots+a_1)$$$$=a_{98}-(a_{97}-a_{96})-\dots-(a_3-a_2)-a_1\le a_{98}\le A,$$ proving the first half of the claim.
To prove the second half, we first show that, after every pair of bags is thrown, the the pile with more beets will have a "pivotal bag," one whose removal will cause this pile to have at most as many beets as the other. Assume by induction this is true after $2n-2$ bags have been thrown, and let's say that Left has at least as many beets as Right at this point.
If Left still has at least as many beets after throwing bags number $2n-1$ and $2n$, then whatever bag was pivotal before is still pivotal now.
If, on the other hand, throwing these two bags causes Right to have more beets, then the bag thrown into Right is now pivotal.
By induction, there is always a pivotal bag. Therefore, the discrepancy is at most the number of beets in the pivotal bag, which is at most $B$, proving the claim.
Now, we have two piles, Left and Right, each with $49$ bags, and the two bags set aside at the beginning. Throw those two bags into whichever of Left and Right (let's say it was Left) has more cakes. Left might have had fewer apples than right, but from the claim, we know it was losing by at most $A$. After the bag with $A$ apples is thrown in, Left now has at least as many apples as right. Same for the beets. Thus, Left now has at least half of each foodstuff.