Is it possible to move podcasts from iPad device to iCloud storage?
I see a number of possibilities here, but...

Podcasts typically don't take a lot space 

...it may be worth mentioning up front that most audio podcasts don't actually take up a lot of space. Typically they use less than 1 MB per minute of audio, so you may not really have an issue (unless you intend to download a lot of episodes from multiple podcasts). 

I listen to a lot of podcasts, so for reference purposes below are some examples of the amount of space taken up by those podcasts:


46 mins (37.2 MB)
40 mins (31.4 MB)
31 mins (24.7 MB)
22 mins (19.9 MB)
51 mins (35.7 MB)
1 hr 3 mins (43.6 MB)
5 hrs 53 mins (243.5 MB)


Of course, there are exceptions. I do listen to a couple of podcasts that actually use up about 2 MB per minute, but in my experience this is rare. And, if you're going to be watching video podcasts these will take significantly more space.

Streaming

The other consideration is whether you're intending to store them for long-term use? If not, then you don't have to download them in the traditional sense. Instead, you can just stream them when you're listening to them, in which case there's no need for you to manually manage your storage (well, at least not with podcasts). 

Of course, in the case of streaming, you do need to have an active internet connection while listening to them - so if that's not going to be possible then this option isn't really going to work. This option also means you're not storing the podcasts for access in future.

So, the point of all this is if you're only subscribing to a handful of podcasts you may not end up using a lot of your iPad's storage. On the other hand, if you're wanting to hold on to past episodes then these can add up quickly! 

For more information on downloading and/or streaming podcasts refer to Subscribe and listen to podcasts.

Podcasts app settings

The bundled Podcasts app actually offers quite a few options to help you manage your podcasts (and therefore the amount of space they take).

For example, you can set it to delete old downloaded episodes after you have listened to them. Likewise, you can set episode limits so you don't accumulate a lot of episodes of each podcast.

However, the trick here is that these settings are managed from iOS Settings, not from the Podcasts app itself. To see these and other Podcasts settings:


Open the Settings app
Swipe up until you see the Podcasts icon
Tap on Podcasts to view the various options available


For example, in your case you could set:


your podcasts to refresh manually
to not download episodes
to delete played episodes


3rd party apps

Apple's Podcasts app works pretty well, but as yet you can't use it to store podcasts in iCloud, you may want to consider 3rd party options.

A number of apps offer the functionality to use cloud storage for your podcasts, mp3s, music, etc.  One example is the VOX – MP3 &amp; FLAC Music Player app. However, while this is free, the VOX Music Cloud is only available for premium subscribers. There would be many other options though, so it would be worth searching around.

Use a desktop version of iTunes

I understand this is not your preferred option, but you can use iTunes on your Mac or PC to subscribe to and/or download your podcasts and then sync them across to your iPad if/when you need to listen to them.

Once downloaded on your desktop computer, there's nothing stopping you transferring the podcasts to iCloud, Dropbox, etc for online storage. Or, for that matter, to an external drive. In my case, my entire iTunes library is stored on an external drive, so any music, podcasts, etc are stored there.