How bad is it to stare at your computer screen all day long?

Not. Staring at a computer screen all day long can exacerbate eye dryness, induce eye strain, possible worsening of myopia in a young person due to constant accommodative state, but shouldn't cause vision loss.
Not so bad. Staring at anything with reduced blinking can be uncomfortable especially with dry eyes. Also, if you need glasses blurred vision can give you headaches. There is no eye damage that comes from looking at your screen but it's good to take regular rest breaks.