There are a couple ways to do this! One is with ImageSize, and one is with RasterSize (probably recommended), and they have slightly different effects.
Specifying ImageSize essentially scales up your graphics image first, and then rasterizes it for the PNG format. Since these lines in Mathematica have a width that is independent of the size of the image they're in, increasing the ImageSize will not increase the line width, and so the lines will appear relatively thinner.
It's possible that's what you want, but more likely, you want to have a high-resolution version of the graphic as you see it! You could potentially "fix" the above problem by adding an explicit Thickness[0.0025], (or some other number) in front of the Line expression (which might be useful anyway, as it lets you control line thickness).
Or, you can simply use RasterSize instead, which specifies how many pixels to break down the image into. RasterSize can be specified in a couple different ways (see the Details section), but the easiest is probably by simply specifying the horizontal width in pixels as an integer. So, defining graphic via your code:
raylengths = {2, 10};
graphic = Graphics[{Disk[{0, 0}, 1, {0, Pi}],
{Dashing[Riffle[RandomReal[.1, 25], RandomReal[.02, 25]]],
Line[{{0, 0}, (Last[raylengths = RotateLeft[raylengths]] /.
2 -> RandomReal[{2, 3}]) Through[{Cos, Sin}@#]}]} & /@
Subdivide[0, Pi, 60]},
PlotRange -> {{-3/2, 3/2}, {0, 4}},
Axes -> {True, False},
AxesStyle -> Directive[Thick, Black],
Ticks -> None];
Export["minimalSun4K.png", graphic, RasterSize -> 3000]
This will give an image with a pixel width of 3000 (and thus a height of 4000).
For the sake of example, I'm using png, but there are other image formats you could use, e.g. .tiff, and I believe the same RasterSize option works for them as well (though I could be wrong about some of them).
To get it to be the right dimensions, you need to either fiddle with the value of PlotRange, use a maximum height in RasterSize, or crop it post-exporting.
To use PlotRange, you might want something like {{-3/2,3/2},{0,3*pixelheight/pixelwidth}}, or {{-d*3/2,d*3/2},{0,d*3*pixelheight/pixelwidth}} where d is a value around 1 that lets you fudge the "zoom" of the image.
I'm not exactly sure whether A4 is 297/210 or 11.75/8.25, but if A4ratio = A4height/A4width, you could use it in the same way: PlotRange -> {{-3/2,3/2}, {0, 3*A4ratio}}.
However, this code uses Axes to produce the horizon, and that produces some artifacts in the bottom corners.
So, instead, I recommend
raylengths = {2, 10};
graphic = Graphics[{Disk[{0, 0}, 1, {0, Pi}],
{Thickness[0.015], InfiniteLine[{{-1, 0}, {1, 0}}]},
{Dashing[Riffle[RandomReal[.1, 25], RandomReal[.02, 25]]],
Line[{{0, 0}, (Last[raylengths = RotateLeft[raylengths]] /.
2 -> RandomReal[{2, 3}]) Through[{Cos, Sin}@#]}]} & /@
Subdivide[0, Pi, 60]},
PlotRange -> {{-3/2, 3/2}, {0, 4}}];
Export["minimalSun4K.png", graphic, RasterSize -> 3000]
where you can change the number in Thickness to your liking.
Let me know if there's any other way I can help!
