Yes, you can say something like...
"She just washed the dishes"
...but what exactly you mean is entirely context-dependent - you could continue with either of...
1: ...but she didn't dry them and put them away.
2: ...a few minutes ago.
Outside of context there's not normally any way to establish whether just means only a little while ago or only that and nothing else.
Regarding OP's secondary question about the meaning being affected by word position...
"She washed just the dishes"
...could only mean only/nothing but the dishes (she didn't wash the pots and pans). But so far as I can see this doesn't lead us to any useful general principle, given that...
3: "I just found out this morning."
and
4: "I found out just this morning."
...effectively mean exactly the same in most contexts (contrary to what might have been expected, I didn't know [whatever] until this morning). But there is a case for saying #3 is more suitable when emphasising that [whatever] is some piece of trivial information you happened to come across, and #4 fits better when you want to emphasise how recently you discovered whatever it was.
To summarise: for any given context, the position of "just" might affect meaning (subtly or decisively), but it's not obvious to me this leads to any useful general principles. In essence, "context is everything".
Best Answer
The reason that anything works here is because, although the sentence is technically positive polarity, it has a type of negative meaning.
Words like anything, ever, at all, and yet usually occur in negative sentences:
However, they can also occur in other sentences which have some type of negative meaning. For example, yes/no questions represent a choice between a positive and negative version of the proposition. So these work fine in yes/no questions:
These types of word are often known as ɴᴇɢᴀᴛɪᴠᴇ ᴘᴏʟᴀʀɪᴛʏ ɪᴛᴇᴍs. The environments they occur in are sometimes known as ɴᴇɢᴀᴛɪᴠᴇ ᴛʀɪɢɢᴇʀs.
The word only, which we see in the Original Poster's example, has a type of hidden negative meaning. Consider:
This sentence means the same as:
And because of the negative meaning involved, we can use negative polarity items such as ever here:
The word only is therefore often a negative trigger, allowing the use of words such as any, every and so forth.
The Original Poster's sentence is:
This means the same as:
This hidden negative meaning makes the use of anything possible here. Notice we could easily use other negative polarity items here too, for example ever or at all: