- They are already extremely loyal to their master(s);
- They are certainly capable of telling the secret to their master(s) after the charm is cast and are almost completely unable to betray them;
- They rarely if ever leave the house;
- No one would ever consider them a potential Secret Keeper;
Here's what we know about House-elves:
I believe, if a house-elf could be made into a Secret Keeper, it would not be a good idea for the elf to be its master's Secret-Keeper, as its master could order the elf to divulge the secret and I don't think the house-elf could refuse.
The Fidelius Charm is complex:
‘An immensely complex spell,’ he said squeakily, ‘involving the magical concealment of a secret inside a single, living soul. The information is hidden inside the chosen person, or Secret-Keeper, and is henceforth impossible to find – unless, of course, the Secret-Keeper chooses to divulge it. As long as the Secret-Keeper refused to speak, You-Know-Who could search the village where Lily and James were staying for years and never find them, not even if he had his nose pressed against their sitting-room window!’
Prisoner of Azkaban - page 153 - Bloomsbury - chapter 10, The Marauder's Map
A single, living soul suggests just a soul; it doesn't say the soul has to be human. Conversely, inside the chosen person suggests that being a Secret-Keeper is specifically a human ability.
House-elf magic is different than wizarding magic. For example, house-elves can Apparate within Hogwarts; human witches and wizards cannot. We don't know enough about house-elf magic to adequately assess a house-elf's ability to be a Secret-Keeper; however, their profound loyalty toward their master, whether voluntary or indentured, does demonstrate a characteristic that would be conducive to being a good and strong Secret-Keeper.
However, what I noted above shows why we don't have enough canon information to say yes or no to this question; there is no information in canon that, when put together in good faith or the spirit of canon, suggests a definitive answer.
the latin root for fedilius translates to trust, as long as someone trusts a being that can speak or write.
in one way one may trust an elf to follow orders and keep a secret but they are part of a race that must do so, this in essence breaks the trust necessary for the enchantment to draw power from.
so in the end it depends on how the spell recognizes trust.
one can assume that dumbledore dumbed down the explanation for harry and he explained it this way.
the way I understand it is it is a perfectly faulted system.
the defenses do not allow the house to be found the secret keeper must tell the person looking for it, this could be through spoken or written word.
now the only weakness is the secret keeper. you must trust him to take proper precautions, to not tell any one and to protect himself.
It is assumed in many cases that the secret keeper cannot stay within the enchanted property as you do not trust him to take care of himself.
It is perhaps implied that they can be.
In Chapter of Order of the Phoenix we have the following statement:
“Oh yes,” said Dumbledore. “You see, Kreacher was not able to betray us totally. He is not Secret-Keeper for the Order, he could not give the Malfoys our whereabouts or tell them any of the Order’s confidential plans that he had been forbidden to reveal.
Saying that Kreacher is not the Secret-Keeper might imply that he just happens not to be the Secret-Keeper but in theory he could have been the Secret-Keeper.