The biggest thing you're doing wrong is ignoring the units when you're working with the quantities.
Now let's look at the rest of the problem:
The formula you used is correct:
Net flux through the surface = (net charge inside) / ε₀
and ε₀ = 8.85 x 10⁻¹² farad/meter.
What's the net charge inside the surface in this problem ?
It's (5.85 x 10⁷ electrons) x (the charge on each electron)
= (5.85 x 10⁷ electrons) x (-1.6 x 10⁻¹⁹ coulomb/electron)
= -9.36 x 10⁻¹² coulomb .
Finally, (net charge inside) / ε₀
= (-9.36 x 10⁻¹² coulomb) / (8.85 x 10⁻¹² farad/meter)
= -1.058 newton-m²/coulomb .
The sign and the significant figures in your answer are correct, so we can see that you know what you're doing. The only thing left is the order of magnitude. You most likely took one of the negative exponents and made it positive. You got an answer that's 10²² too small. Big deal. You could claim "that's close", and see whether you can convince a teacher.
