A hyperelliptic curve over Q is called "locally soluble" if it has a point over every completion of Q. In this paper, we prove that a positive proportion of hyperelliptic curves over Q of genus g≥1 are locally soluble but have no points over any odd degree extension of Q. We also obtain a number of related results. For example, we prove that for any fixed odd integer k>0, the proportion of locally soluble hyperelliptic curves over Q of genus g having no points over any odd degree extension of Q of degree at most k tends to 1 as g tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer-Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to 2, together with suitable arguments from the geometry of numbers.