Medieval Europeans knew both that the earth was round and that the sun's course around it varied to create different lengths of day and night, from which they could readily infer (with varying degrees of accuracy) that there would be places which would have constant day or night. So, for example, Bede explains how the length of days and nights vary in summer and winter to an opposite and corresponding degree in the northern and southern hemispheres:

The reason why the same [calendar] days are of unequal length is the roundness of the Earth [...] The Earth being thus shaped and given to mortals as their habitation, the orbit of the Sun, which is always shining in this universe, gives daylight to one place and leaves another in night by the firm decree of God’s law. [...] Again, it is necessary that the fiery Sun mount up the middle orb of heaven at one and the same point in time through the whole circle of the year for all who are placed facing one another in the same [vertical] line under the northern or the southern clime. However, it does not rise and set at the same moment or hour for both, but travelling through the southern zone in wintertime, it rises earlier and sets later for those who inhabit the southern regions of Earth than [it does] for us who, placed towards the north, with the globe of the earth blocking the way, receive its rising later, and its setting earlier. But on the other hand, in summer the Sun rises very much earlier for us who live under the same latitude, and seems to remain much longer on the point of setting [with us] than with those who dwell on Earth’s southern flank. They are denied an earlier sight of [the Sun] by the imposition of the Earth, and then are obliged to renounce it sooner. Therefore they have shorter days in summer than we do, but longer ones in winter.

He also notes the classical discussion of the quasi-mythical island of Thule, the furthest northern point of the inhabitable world:

In discussing these matters concerning the varying length of the solstitial days, Pliny leaves the length of the winter [solstitial] days in those regions to be inferred. But he makes it equally clear what the length of the night in both seasons is, because it is necessary that the days, be they of whatever length, make up, together with the night, a space of 24 hours. But it should be noted that Solinus writes these things about Thule in another vein: In uttermost Thule, when the Sun passes at the summer solstice through the star Cancer, there is no night, and likewise no day at the winter solstice. Nor does Pliny contradict this in his seventh book:The uttermost place of all they call Thule, in which we have indicated that there are no nights at the solstice when the Sun passes through the sign of Cancer, and on the other hand no days at the winter [solstice]. Some are of the opinion that this happens for six months.

(Bede, On the Reckoning of Time, 31-2 (trans. Wallis, 91-2).)

This idea that Thule had no night in the summer and no day in the winter was very well known. Not only was Solinus, as well as Bede, fairly widely read, but Isidore of Seville also mentions it in the Etymologies (probably the most widespread encyclopedia through the middle ages):

Ultima Thule (Thyle ultima) is an island of the Ocean in the northwestern region, beyond Britannia, taking its name from the sun, because there the sun makes its summer solstice, and there is no daylight beyond (ultra) this. Hence its sea is sluggish and frozen.

(Isidore, Etymologies, 14.6.4 (trans. Barney et al., 294).)

Although some authors interpret this to mean that Thule has 6 months of constant day and 6 months of constant night. For example, Honorius Augustodunensis in his widely read encyclopedia, the Imago mundi, describes Thule thus:

[There is also] Thule, the trees of which never loose their leaves, and in which for 6 months, viz. in summer, there is continual day, [and] for 6 [months] in winter [there is] continual night. Beyond this to the north the sea is frozen and [it is] perpetually cold.

Thile cuius arbores numquam folia deponunt, et in qua .vi. mensibus videlicet estivis est continuus dies, .vi. hiberis continua nox. Ultra hanc versus aquilonem est mare congelatum et frigus perpetuum.

(Honorius, Imago mundi, 1.29 (ed. Flint, 62-3).)

Follow on question:

How did North American and Northern Asian peoples interpret the elongated days and nights?