As Suki has said, English does have quite a lot of words from Old English, it’s just that they tend not to be the most interesting ones. Actually, a lot of profanity is from Old English, but that would go beyond the scope of this post.
Felipe is correct in noting a fundamental difference between English and Portuguese in this respect. Languages like Portuguese and French are descended from Latin, and the vast majority of Portuguese and French words are of Latin origin, or Greek words that had been borrowed into Latin. English, on the other hand, currently has over 1/4 of her present vocabulary from Latin, and another 1/4 from French (which is essentially Latin as well). More recently, English has continued this practice.
I believe the willingness of English speakers to “borrow” words from other languages is a great part of what makes English so rich. English doesn’t even have an official language authority. It may be amusing to contrast this to the Royal Academy of the French Language, whose members wear funny green uniforms, carry swords, and are known as “the immortals.” Just in case you think I’m joking: