There is keyboard shortcut in gnome that turns the screen reader on or off, and it is AltSupers. (Super is the windows key)
To see this and other accessibility shortcuts:
Open the Settings app

From the side bar, navigate to the "keyboard" section

Scroll down to the Keyboard Shortcuts part, and click "View and Customize Shortcuts"

Here, you can see and modify all the keyboard shortcuts. For the screen reader, click Accessibility:

Here is the screen reader shortcut:

You can also toggle the screen reader in gnome via the terminal, by running
gsettings set org.gnome.desktop.a11y.applications screen-reader-enabled true to turn it on,
or
gsettings set org.gnome.desktop.a11y.applications screen-reader-enabled false to turn it off.
The Orca documentation says that you can also start Orca with screen reader by running orca -e speech in your terminal or the Run dialog. Although, when using the terminal, I had to resort to Controlc and kill the process in order to stop it, and when started from the Run dialog, I had to go to the Accessibility section of the settings menu, and then toggle screen reader on and then off once to stop it.
If it is useful to you, you can bring up the Run dialog by pressing AltF2:

You might also find some answers to How do I stop Orca Screen Reader?, mainly this and this one, helpful.