You can use the passport, but you can't apply in Asmara at the moment. According to the embassy website, the Eritrean government has been very unhelpful of late regarding diplomatic pouches, so the embassy has been closed for visa applications since January. You can check this out for yourself at http://eritrea.usembassy.gov/questions_and_answers.html
So, if you can get to Kenya or some other country, you could apply. However, there is no guarantee that you would be issued a tourist visa. It's usually easier to get a visa in your home country. It's harder to convince an officer in another country that you intend to return.
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