I think any normal person, when asked how many numbers there are, would probably say, "plenty" – or something similar. I imagine a few people would even mention that it's infinite. But even after some basic math courses in college, it's possible to not be aware that there are far too many numbers to even begin to imagine.
To clarify: the smallest number of infinite things you can have is basically things you can count off, if you had all the time in the world. This is a pretty crowded category, as you might imagine – most things are infinite the same way, to the point that saying "there are different, bigger infinities" isn't something most people are primed to properly understand. Suffice it to say, everything you can imagine is (very likely) only just barely infinite – what fancy-pants mathies refer to as "countably infinite".
Not just integers (like -1, 0, 1, 2, 3, 4...), which you can count off by doing the negatives after the positives and counting up normally. Not just fractions (13/27, 7/3, 14/5, ...), which you can sort of imagine by counting off all the fractions where the numbers are all less than a certain size, and then just slowly getting bigger sized numbers (as long as you skip duplicates like 2/4 and 3/6). But also all the numbers you can get a computer to calculate – you can imagine the programs calculating them as the binary 1s and 0s, and then instead of running it, you can just treat it like a regular old number and count those – and just ignore all the programs that don't work, obviously – the program for 12 will surely immediately stop after having done nothing or breaking in some bizarre way. But as long as you can count them, they aren't unimaginably infinite. There's just plenty of them, no matter how many you want.
Of course, one familiar bit of mathematics isn't like this. The number line has more points on it than you could ever even begin to count, every tiny segment packing more numbers than a computer could ever even begin to try and calculate or describe with every book you could possibly write for it.
Take, for instance, the busy beaver numbers. The details, while fascinating, are not important – suffice it to say, they describe questions so hard you cannot be truly sure of the answers. The answers exist, of course – they must, the question only asks what is the longest time you can take, and since there are many (but a finite number) of examples, one of them is the record-setter. But if you say, oh, I have a number, it's 0.<the busy beaver numbers written out> – that number exists. You could even point to (about) where it would end up on the number line. And surely, since it exists, and we know just about where, we know the number line doesn't just jump over it. It's continuous. That's the thing we know about lines.
So we routinely use math that requires numbers so deeply strange our merely human devices cannot possibly access them. And the numbers beyond description or calculation aren't just some of the number line – they make up almost all of it.
Of course, you can try and pretend these strange numbers don't exist, but those holes you create in the number line are simply too much – many of the interesting things you do with lines and continuity would be destroyed. We're stuck with this semi-Eldrich knowledge that important things cannot fit inside the human mind (or, indeed, the physical universe).
[p.s. math is fun – not, ya know, as fun as you might guess, reading this, but it's pretty good. In particular, several days after I wrote this (but still several days before publication) numberphile posted a related video – please check it out, both Brady Haran and Matt Parker are gems.]