Yes you can get headaches from using the computer. It can depend on how flat your computer screen is to the refresh capabilities of your graphics card and monitor.
how flat is your monitor screen? Any curve at all makes viewing online information more difficult, no matter what size of the monitor or the resolution settings. The flatter the screen, the better. (These days, monitor screens come very flat...but at a cost, of course.)
Also, what refresh capabilities and type do your graphics card and monitor offer? Monitors actually display things onscreen with an electron beam directed from the back of the monitor to the screen itself (very similar to a projection TV). The beam starts at the left end of the screen, draws across the screen, goes back to the other end, draws another line, and so on (like a typewriter).
The refresh rate, then, refers to how many times per second the entire screen is drawn one time. How many trips across the screen the electron beam must make depends on the resolution. That is, if you have 640X480, then the beam makes 480 trips across the screen to refresh it one time. If you have 800X600 resolution, the beam will make 600 trips. And so on.
For most of us, we don't consciously detect the screen refreshing. But, while we see a full screen, our brains are actually filling in the "missing" lines for us. If you take a photograph (at a high shutter speed) of a monitor that's using a fairly low refresh rate (say, 60Hz), you will often see only half a screen drawn; the other half will appear black...yes, because it's not really there. Our brains fill in the missing lines. Also, if you see a computer monitor on TV, you can often see the screen refreshing because the scan rate of the TV is much less than monitor rates; the refreshing process appears as lines moving from top to bottom. Again, while we're sitting at the computer, the refresh process is fairly undetectable to us, but our brains must work to compensate for the screen as it refreshes.
Complicating, the refresh process is that it can be interlaced or non-interlaced (just like .gif graphics!). If it's interlaced, then the electron beam scans rows 1, 3, 5, 7, etc., often creating an undetectable-to-you flicker. Like the refresh process itself, you may not notice the flicker, but your brain will, making it work even harder to fill in missing information. Non-interlaced refresh draws rows 1, 2, 3, etc., and causes much less flicker for your brain to detect and overcome.
So, back to computer-induced headaches.... You need to have both a monitor and a graphics card that support a refresh rate higher than 70Hz and a non-interlaced refresh process. Also, you can improve the display by experimenting with your display settings. In Windows 95, 98, or NT, go to Start-->Settings-->Control Panel-->Display-->Settings-->Advanced-->... At the bottom of the dialog box (assuming your graphics card provides options), you can specify a refresh rate setting. Note that the "Optimal" option is often not...well...optimal, and you'll likely have better performance by setting a specific refresh rate.
To experiment, start by setting the refresh rate as high as it will go at the resolution you're using, then start working down in the resolution settings. In general, you'll likely have a better display using refresh settings that do not max out your graphics card and monitor capabilities. For example, if you're using 1024X768 resolution, your graphics card will need to work much harder to refresh at a higher rate (because there's more lines to refresh than at lower resolutions). Often, setting a slightly lower rate will result in better onscreen display because the card is working comfortably within its capabilities. Experiment with settings to see what display settings work best for you.