It appears you are using a "plain" M42 adapter, i.e. a piece of machined metal without any electronic components that enables the M42 lens to screw onto the Sony camera body, e.g. like this one (SKU: M42-SNA-P-V1):

This is fine for most cases. But, as the plain adapter does not communicate anything to the camera, the camera thinks there is no lens attached. Therefore, for this to work you must be able to instruct your camera that it is safe to fire the shutter without any lens attached.

As osullic writes in their comment the Sony A350 appears not to have such a release without lens override.

Per user24582's suggestion, you need to look at an adapter that has a small electronic "AF confirmation" chip attached onto it (note the strip of contacts in the upper left):

What this chip does is two-fold:

• It tells your camera that there is a lens attached.
  • The characteristics of the attached M42 lens (focal length, aperture value) are not known to the chip (as the M42 lens cannot communicate those), so the chip always returns a pre-programmed value.
  • My (EOS) M42 adapter for example always makes it appear as if a 50 mm lens is attached with its aperture set to 1.4.
• It allows your camera's AF feedback mechanism to work (partially).
  • The M42 lens has no motor and cannot be driven by the AF motor of the camera body, so you cannot fully autofocus the lens.
  • However, by manually turning the focus ring while half-pressing the shutter button (or whatever button you've set for the AF) you will get a confirmation (beep, or blinking light) when the scene is in focus.
  • Personally I found this a very useful feature, as without a split-prism focusing screen I found it hard to focus manually using a DSLR.