No, the correct answer to the correct question (where you added in "every year" for clarity) is not either of those you state, but $25,000/year.
You wrote a confusing question, by mentioning the 10 years but asking about the annual amount. If the question were, "What is the total amount he will get?", then your answer of $250,000 would be correct. But you asked "How much would he receive annually?", so the answer has to be in $/year, and does not involve the 10 years. It is just 1/4 * $100,000/year = $25,000/year. The units cancel just fine either way.
Just to keep this from being a short answer, let me state the whole thing again:
The data given are:
An executive receives a severance pay of one fourth of his annual salary of $100,000.00 every year for 10 years.
The natural question that could be asked is:
How much would he receive (total) over the 10 years?
The answer to
that is:
1/4 * $100,000/year * 10 years = $250,000
The question you posed is different:
How much would he receive annually?
For this, we don't care how long he is being given the money; "annually" means
each year, not
total. So the answer is:
1/4 * $100,000/year = $25,000/year