Can I withdraw from a 403b without having a hardship?

Generally, withdrawing non-qualified distributions from retirement plans is a very expensive way to raise cash, and will trigger a very large tax hit as you've noticed (your regular income tax on the amount withdrawn + additional 10% penalty).

Most plans don't make it easy, and most don't allow withdrawal while in service.

As per the IRS, withdrawals from a 403(b) plan are allowed in these cases:

In addition to loans and hardship distributions, a 403(b) plan may allow employees to take money out of the plan when they:

• reach age 59½;
• have a severance from employment;
• become disabled;
• die; or
• encounter a financial hardship.

If you're still employed with the sponsoring employer, do not have a hardship, are not dead or disabled, and not past the age of 59½, it is very likely that the plan will not allow you to make withdrawals, but you would want to check with the plan administrator.