Are there any safety issues with this breaker box?

• The RV circuit ought to have a ground wire to the panel. It having a local ground rod means grounded metal parts of the RV won't float up to dangerous voltage levels relative to the surrounding dirt, but without a ground wire back to the panel various faults to ground might not trip the breaker. Grounding might not have been required at the time of installation, but if you have the ground wire back to the panel there's really no good reason not to connect it.
• The RV circuit probably ought to have GFCI protection, given that it's outdoors and especially if you fix the circuit's grounding.
• The RV circuit cable must be clamped, not loose in the knockout. It could wear against the sharp edge of the knockout and cause a short, arcing, or fire.
• You probably have “alien” breakers. All breakers must be specifically listed or classified for use in your specific panel. You may not mix and match at whim. Some types may seem to fit, but may lead to arcing, bus damage, or fire. It looks like there are at least three brands present (Siemens, Square-D Homeline, something else) and so some of them are probably not appropriate. Your panel looks like a GE unit based on the model number starting with "TLM".
• It sounds like you said you have no shut-off between your meter and this panel. You need one, to protect the feed wiring and panel bus. It can be legal to have a panel with six breakers and no single main, but you have seven (and room for more).
• The feed wires look a bit small for the total loads, especially if there’s no main breaker protecting them.
• The wire to the ground rod needs to be the full size required by code (#6 or #8 depending on feed size) all the way to the bus bar lug, not spliced down to a smaller gauge (looks like #10 at most) for any length.
• There are five neutral wires coming in the right side of the panel but only four hots. What’s the extra for? If it’s for a disused circuit, it should be disconnected from the bus bar.

Adding to another answer:

Looks like the neutral is quite a bit larger than the feeders. When I see that I think they just wired it with "stuff they had laying around".

There are specific rules about "taps" (NEC 240.2): Tap runs of 10' or less must be enclosed in a raceway, have an ampacity of at least 10% of the circuit it's attached to and terminate with an over-current protection device appropriately sized for the feeder wires. I'm not sure how this applies when there is no real rating for the supply circuit (from the meter, if I read the OP post correctly).

At a minimum the OP needs better overcurrent protection for that panel. He has space, so he could simply get an appropriately rated 240 volt breaker and backfeed it to supply the panel, then it would become his main breaker and comply with overcurrent protection. Unfortunately, that probably means pulling the meter to shut of power, given the way it's connected. And depending upon the POCO, it may lead to pulling a permit, and inspections, maybe bringing more and more stuff up to current code. Could snowball.

Do you planning first. Check if the POCO will pull and put back your meter so you could add a main breaker without involving a permit or inspections, etc. If so, that would be my first priority to make it safer. That would be a baby step. If you really have an ungrounded, fuse type main panel, it might make more sense to take a more holistic approach for your upgrades, could be less expensive and much better outcome in the long run rather than doing it piecemeal.

A ground rod is no substitute for a ground wire. They do completely different jobs. The RV branch needs a ground wire in addition to the ground rod. Since the cable is orange, clearly it was manufactured with a ground wire in it; that will have to be restored by either putting it to good use, or replacing the defective parts of the cable if it is unuseable.

The orange cable needs a cable clamp fitted. You'll need to pull the cable through it, so that'll be a good time to fix the ground issue.

I'm not worried about the neutral wire being bigger; bigger is always allowed, and going to #4 or larger allows remarking a black wire to be a neutral, which is a clever hack. What greatly concerns me is that the wire run is completely unfused. You need to either feed this from the main panel, or change the meter to a "meter-main" with a few auxiliary breaker spaces, and fit a 60A(?) breaker in the auxiliary spaces to power this sub. (breaker amperage as appropriate for hot wires; in some cases 70A is OK for #6). I would fit the meter-main first; that way you can de-energize your various panels. That'll make them a lot easier to work on.

The dog's breakfast of alien (Square D, Siemens) breakers need to be changed for ones correct for the panel bus bars. Continued use of alien breakers may result in bus stabs being burned up and possible fire.

Double-tapped grounds are usually OK if the panel labeling permits that.

You'll need to clean up this panel and fit a main breaker + retainer to it

Your first order of business is to clean this mess of a panel up, and that's a two-pronged job. First off, you need to remove the existing breaker from the top left and fit a THQL2160 main breaker with a THQLRK1 retainer kit there, then, with the utility having cut power at the meter, move the existing hot wires to the new backfed main breaker.

Once that's addressed, and the service barriers that come with the retainer kit fitted, we can then move on to cleaning up the alien breakers that have infested this panel. Each and every breaker in this panel that isn't either a GE THQL or an Eaton CL will need to be replaced with its THQL counterpart (same poles, same amp rating).

And fix that dangerous floating RV ground

Your RV outlet needs a grounding wire run from it to this panel, no questions asked, as that ground rod is doing nothing for you in terms of protecting you from wayward utility power. Given that this is a shop/shed space, and there's a ground rod at the RV outlet already, I'd just run a #8 Bare Armored Ground cable from the RV outlet box, into the wall, and onward to the breaker box, extending the wire inside to terminate on the bar there, provided the RV box is on a wall of the shed and not on a post outside.