Is a severance package with an attached non-compete clause legal?

Generally, you can't get something for nothing. If it's part of your employer's regular policy to offer severance pay, your employer is not "sweetening the pot" in any way and you have no obligation to agree to the non-compete clause.

However, courts tend not to like non-compete clauses even under the best of circumstances because they're designed to keep you out of work in the field you know best. Unless you're being offered substantial sums of money and the non-compete is very narrowly drawn, it's questionable whether a court would uphold it. But you don't want to have to wait for a court to rule on the situation before you know. And different states have different degrees of tolerance for them (California tends to be pro-employee, so you're fortunate there).

Have an employment law specialist in your area (find one who focuses on the employee, not management/employer, side) review the non-compete agreement with you.