Can I install required breaker in my electrical panel?

Can you install a new 240V breaker? Probably. It looks like the necessary two spaces are available on the bottom-right (the one space on the left side can only get you 120V without shuffling things around).

Pictures of that areas straight-on would be helpful, as would pictures of the panel front and the stickers on the inside left and right sides of the box.

Since there is no shut-off in this panel, it must be somewhere upstream - probably in the same box as your electric meter. You'll want to take a look and see what the amp rating of your feed is, as that could limit whether you could run this new load at the same time as all your other existing loads. Judging by the wires feeding the bus bars, you probably have 100A service to this panel.

For an outdoor receptacle, you'll need GFCI protection somewhere in the circuit - either in the breaker or the receptacle. Receptacle GFCI can be less expensive than breaker GFCI, but may not be available in the size you want. It's also not a great idea to mount sensitive GFCI electronics outdoors.

There are some other concerning issues in your picture:

• No grommets/clamps around any of the cables entering the box. Without proper clamps, the cables may rub against the sharp edges of the knockouts until the insulation is cut and a short circuit occurs.

• Probable missing handle tie on the breakers in spaces 5 and 7 (left side, below the double 30). Judging by the lack of visible numbers on those handles and the black and white load wires, those two are probably feeding one 240V load and used to have a handle tie. You don't want to be able to turn off only one leg to the load, as it creates a dangerous situation where the device appears to be off but still has live wires inside. A handle tie ensures you always turn off both legs together.

• Probable missing handle tie on the breakers in spaces 15 and 17. Also, the wire is far too small to normally be on a 50A breaker (it looks like probably #12, good for 20A or maybe #10, good for 30A), although there are code exceptions for certain motor loads (such as air conditioners) that mean it could be a legitimate configuration.

• At least two of the screws in the neutral bar have two wires. There should only be one neutral wire per screw. Grounds can generally be doubled up.

• The orange cable should not be flopping out in the open air. It should be fastened securely, and protected from damage (e.g. covered by wall or conduit below a certain height, which it is surely below if the panel is mounted at a legal height).

• The ground wire in the bottom-left corner of the panel (the one mostly wrapped in paper) looks like it isn't connected to anything.