Yes, it is possible to read texts that are written only in pinyin. This is pretty trivial in one sense: pinyin spelling indicates all of the segmental phonemic distinctions of standard Putonghua Chinese (it was designed to) and when used with tone marks and correct word division and punctuation, it indicates some of the suprasegmental and intonational aspects of pronunciation as well. So someone with a pinyin text can read the sounds out loud fairly well, and if you can do that, you should be able to understand what's being said about as well as you can understand something that you hear that is not written down.
In another sense, it's obviously difficult for people who are used to one spelling system to read something in an unfamiliar spelling system. /fɔr ɪgˈzæmpəl, aɪm ˈraɪtɪŋ ðɪs ˈsɛntəns ɪn ði ɪntərˈnæʃənəl fəˈnɛtɪk ˈælfəbɛt; ɪts ˈprɑbəbli ˈhɑrdər tə rid ðən ðə nɛkst ˈsɛntəns./ For example, this sentence is probably easier to read than the preceding sentence written using the International Phonetic Alphabet. But it is not impossible to read an IPA transcription of English; it just requires knowledge of the sound correspondences, and practice if you want to be able to read quickly. So this is a practical objection to replacing characters in pinyin; it doesn't mean writing comprehensible Chinese in pinyin is "absolutely impossible" in principle.
And in another sense, a non-phonemic writing system allows you to get away with things that would be confusing, or even incomprehensible, in spoken language (like the "shi" poem). I found a tract calling for English spelling reform that gives the following English example:
the words "attend to know understanding" might easily be read to
convey the meaning "attend to no understanding"
A plea for phonetic spelling: or, The necessity of orthographic reform, by Alexander John Ellis
But as Ellis points out, such sentences can simply be reworded to make them less ambiguous in their spoken form. Furthermore, this isn't something that's only possible in post-reform spelling systems. Many native English speakers are familiar with sentences such as "Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo" or "James while John had had had had had had had had had had had a better effect on the teacher" that are technically grammatically "correct" but impossible to understand even when written down in the current English spelling system. This could be seen as a "flaw" of the current English spelling system, but in practice it is not a problem for English speakers because nobody needs to use sentences like this anyway.
Similarly, no one actually needs to be able to write sentences or texts composed entirely of variations on the syllable "shi" in Chinese (I say "variations" because the syllables are not actually completely identical; there are different tones which can be indicated in Pinyin with tone diacritics or numbers).
The "shi" poem is an extreme example, of course, and doesn't represent the real issues that would arise with spelling reform.
The tendency to ignore the potential of confusing homophony when writing in characters may make it difficult to read some texts that were originally written in characters that are re-written in pinyin. (Furthermore, as michau points out, it means that some established specialized or technical terminology is currently disambiguated mainly by the spelling; something like a more extreme version of the distinctions in English between phonetically similar words like "prescribe" and "proscribe", or "alkanes", "alkenes" and "alkynes", or "hypothermia" and "hyperthermia".) This doesn't make it impossible to write texts in pinyin that are understandable. But, it does mean writing Chinese without characters may require some changes in word usage compared to writing Chinese with characters.
However, I think that language reformers in other places have made changes in word usage/vocabulary that are of comparable or greater magnitude. For example, in the transition from Ottoman Turkish (written in a form of the Arabic alphabet) to modern Turkish (written in a form of the Latin alphabet) many words of Arabic origin were more or less replaced with words built on Turkish roots. Likewise, it would not be impossible for Chinese writers to avoid overly ambiguous old vocabulary terms and devise some new standard replacements if it became necessary. Obviously, this kind of vocabulary replacement would have cultural impacts that I am not qualified to evaluate, but there is nothing inherently impossible about it from a linguistic point of view.
The linguist Victor Mair has written a lot about pinyin and characters and how they are used in modern Chinese writing. Here is a blog post he made that mentions the memoirs of Chang Li-ching (his wife) that were published in pinyin: http://languagelog.ldc.upenn.edu/nll/?p=27423 The article and comments mention that many native Chinese speakers find it hard to read the text fluently, but it is evidently not literally "impossible".
As many of the other posts on this page have mentioned, the details of standardized writing systems are more strongly influenced by political and cultural considerations than by whether it is literally "possible" or "impossible" from a strictly linguistic perspective to use any particular system for any particular language.