ArcGIS Online and Sign In options greyed out in ArcMap?

On your Windows Task Bar, near the clock, you should see a globe icon (you may need to choose the Show Hidden Icons arrow).

Right-click on this and choose Test Connection Now - this will attempt to connect to ArcGIS Online. When it's connected, hovering over the icon should show "Connected to ArcGIS Online".

If that fails, try Run Connection Test to debug exactly where it's failing.