Here is the graph:
which has rounded corners when you look closely:
How do I sketch this?
My thoughts are as $-1 \le x,y \le 1$
if $-1 < x < 1$
$x^{1000} \approx 0$
$\therefore |y|^{1000} \approx 1$
By symmetry, the reverse argument applies for $y$.
This gives the "square" shape with the seemingly straight "sides".
The corners are rounded as we are dealing with a continuous graph so if $|y|$ decreases slightly, then |x| increases slightly to compensate
(reverse argument applies for $x$)
This graph is difficult to sketch without software so any further insights from the community on how to sketch this graph would be much appreciated.
You are asking about a special case of the $p$-norm, where the "unit circle" is defined by $$ |x|^p + |y|^p = 1 $$ for various values of $p$. When $p=2$ it's a circle. As $p$ approaches infinity the circle looks more and more like a square, as you note.
(From wikipedia: https://en.wikipedia.org/wiki/Lp_space)
When $p$ is large it's hard to sketch since at any reasonable scale it looks like a square. You need software to zoom in.
To get a reasonable approximation for $p$ not too large you could use a calculator to solve $$ 2x^p = 1 $$ to find out where the "circle" meets the diagonal line $y=x$. Then you could fill in with a freehand curve and be confident of reasonable accuracy.
It's fun to check this out for values of $p$ between $0$ and $1$.