Box joints for this table with just jigsaw and chisel?
A jigsaw is not the right choice for this. Much preferable to cut this using a hand saw of some kind.
but of course from a stability point of view that is probably one of the worst table designs to come up with ;).
Actually the design is very stable, in all ways you might mean it.
The cantilevering makes it very stable in a gross physical sense. The grain of all parts is in the same orientation so it's stable in terms of seasonal movement. And the box/finger joints have tons of glue surface area so they are inherently very strong if done well. Overall the table is probably a lot stronger than you'd think. Based on a discussion I read about a chair project (yes a chair!) built similarly this should be much stronger than many woodworkers would assume from looking at it.
I'm pretty new to woodworking... If I work carefully will it be possible to make the joints fit well enough to make the table as stable as it needs to be?
We can't answer that for you as it is of course entirely up to you ;-)
But if I can put it this way, there's no reason a relative newbie can't build something like this if they were diligent and didn't rush.
Or would I really need precision machines like a table saw with box joint jig?
A jig and a table saw or router would of course make this a lot simpler, faster and less effort, but needed? No, no more than power tools are needed to produce M&T joints or dovetails.
Any tricks to make this easier?
Practice is actually the thing that'll make this easier for you.
It sounds trite but even hand-cutting dovetails is basically a very simple process at heart yet many struggle to do them well initially for a host of reasons. But once they have tackled enough of them to become reasonably proficient they can find they're actually relatively easy to do, and a real pro can do them amazingly quickly despite the precision that's required for them to fit well.
There are lots of guides online to doing dovetails by hand and you can essentially adopt the process wholesale when doing these, i.e. cut 'pins' or 'tails' first (obviously no actual pins or tails here), then transfer marks to the interlocking piece, so that any irregularities are allowed for.
Some of the following Q&A may be of help: How do experts make dovetails by hand that fit so well?