To begin with you could really do with a try square or similar for marking square. Some form of square1 should be considered standard equipment and many woodworkers find it useful to have more than one, e.g. a try square and/or a combination square for general use and an engineer's square that they might reserve for setting up machines.
I should mention that if you had to or wanted to you could build a try square for yourself, they make a very good early learning project. Perfect accuracy is possible if you use the right methods.
If I just eyeball it I'll often be off by as much as 10 degrees, because it's hard to hold a tool at an exact perpendicular. If I mark off one side with a pencil line, I can cut along the line fine, but the cut is angled by up to 5 degrees to the other side.
Practice, practice, practice. Most of us just don't saw enough to get really good at sawing square by hand and eye, but if you set aside a couple of hours to do nothing but practice square cutting you'll surely get a lot better at it2.
Many of the earlier hand-tool woodworking guides state that most people have a natural tendency to saw to one side or another — away or towards the torso — and I've found that to be the case personally. If you identify that you too have a bias in cutting you can begin to deliberately compensate.
I don't saw enough to get good at it
This is the standard dilemma with leisure woodworkers, we want to be better at various tasks but don't do them enough to get good at them (at least initially). For some tasks the solution to accuracy problems is to jig it.
One simple trick for helping to saw ends square is to use a sawing guide clamped to the board, similar to the chiselling guides shown in this previous Answer.
And there's always the classic mitre box. Paul Sellers has a good video (here) showing how to make your own which is worth a look. Following his instructions if you still struggle to get a perfectly vertical kerf you can always cheat3 and go back to the previous tip and use a square sawing guide.
But surely it is possible to cut a very good perpendicular (error <1 degree) with hand tools only (pencil, straight edge/ruler, hand saw)?
Adding in a square of some kind yes, even cutting freehand.
To be clear, I'm not talking about cutting extremely irregular objects. I'm asking about cutting simple rectangular planks/boards, or long sticks with rectangular profiles (and 1-2" thick). However, as is often the case with cheaper lumber these may not come with perfect corners or edges.
I'm not sure how accurate you want to be or the kind of accuracy you actually need in the work you're currently doing, but do be aware that trying to saw board ends very accurately square first is sort of putting the cart before the horse. Except in carpentry (where very high accuracy isn't needed) it's normally the case that at least one face and one edge are made flat and square first to turn them into datum surfaces, before marking your cuts and sawing.
And for fine work it is normal to use a plane and a shooting board to shoot the ends to make them perfectly square while simultaneously getting the board to final length.
1 A caution on buying squares in general, you unfortunately can't assume they are square! Yeah they have one job but manufacturing these days ain't what it used to be. Woodworking try squares are the most notorious for being out of square and many can't be fixed (easily) so it's well worth buying these in person and checking for square in the shop and getting the best one. Combination squares may be out a bit too, these can sometimes be fixed with a little careful filing inside the groove of the fence. Speed squares, because they're cast in one piece and not machined or assembled from parts, tend to be dead on (including inexpensive plastic ones).
2 Same deal with sawing dovetails, chopping mortises, planing flat and some other things, which benefit from concerted practice rather than the accumulation of experience over a longer period.
3 Note: not actually cheating :-)
