Mars: no magnetosphere required
Just to clarify concepts, the Earth's magnetosphere does not prevent sunrays from being "lethally radioactive" during daytime, but it does protect the whole atmosphere from being eroded, including the ozone in the stratosphere.
Ozone prevents UV rays from reaching the surface and provoking major mutations on DNA sequences. Such mutations do not kill us instantaneously but give us cancer and prevent DNA to be copied adequately to the offspring, so all multicellular life (with long DNA sequences) becomes unviable in few generations.
What you are looking for is a mix between the Earth and Mars. It seems that Mars was a hospitable place some billion years ago, with a warm atmosphere that could hold liquid water and microbial life. However, for several billion years, the solar wind eroded the martian atmosphere so Mars is now a desert. During this process, there must have been an era in which the conditions for life started to disappear slowly.
Your planet could be situated in a similar era, when ozone in the atmosphere is being depleted at a dangerous rate so it is unsafe to expose oneself to sunlight.
PD. I forgot to say that Mars --the same as Venus or the Moon-- does not have a true magnetosphere.
The nature of the interaction between an unmagnetized planet and the supersonic
solar wind is determined principally by the electrical conductivity of the body. If conducting paths exist across the planet’s interior or ionosphere, then electric currents flow through the body and into the solar wind where they create
forces that slow and divert the incident flow. The diverted solar wind flows around a region that is similar to a planetary magnetosphere (p. 523 in the link).