Before addressing your questions, you should understand what a 'preferred share' typically means:
A preferred share is a share in a corporation, which provides the holder with:
(a) probably a fixed redemption amount that must be paid out before common shareholders [that perhaps the shareholder can trigger, perhaps the company can trigger, and perhaps only gets triggered if the company liquidates and closes down];
(b) probably a fixed annual dividend amount that must be paid out before common shareholders [and if no dividends are paid in a year, then they may or may not need to be paid out the following year, depending on terms of the shares]; and
(c) possibly voting rights.
In your case, you've stated that these would be non-voting shares. Note that there must always be at least some shares that have voting rights, otherwise the corporation would have no legal basis to elect its board of directors, and would be functionally something very different than a corporation!
You have a couple of separate questions there, let's look at each one individually:
Would they own the company
Well, they would own some specific amount of value of the company; for example in a company worth $1B, with $300M of issued preferred shares and $700M of issued common shares, if someone held 100% of the preferred shares, they would own $300M of the value of the company, or, 30% of its current value. But because preferred shares have stated redemption values, if the company increases in value by $100M next year, the preferred shares would still be worth $300M, and the common shares would be worth $800M. Transversely, if the company decreased in value by $50M, the preferred shares would still be worth $300M, and the common shares would now be worth $650M.
but have no say in it's direction?
Any 'say' in direction exercised by shareholders, is done solely through using their votes, if any, to vote in the board of directors, who appoint the CEO, who appoints the executives, etc.. If there are no votes attached to your shares, the only reason the company cares at all about you, in theory, is to follow financial regulations which may require certain actions to be taken to protect its shareholders. ie: corporate fraud by the CEO is illegal.
Note also that if the preferred shareholder has the right to redeem the shares at their own discretion, then the shareholder has leverage over the corporation. ie: 'Change the CEO or I will redeem my shares, forcing you to pay me off immediately'. That threat may have power or it may not, depending on how easily the corporation believes it can produce the financing required to pay off the redemption of the shares. So shares that have that right, would have more value than shares that only the corporation can decide to recall.
Are they liable for anything?
The single main concept of what a corporation 'is', is a way to aggregate funding for a venture, separating the liability of the company, from the liability of its shareholders. If a corporation goes bankrupt, no shareholder is liable for debts of the corporation, except in specific circumstances where something like fraud takes place [this is called 'piercing the corporate veil', and refers to instances where regulators may have determined that a corporation was a sham organization].
I guess I'm curious if this has or can happen and what it would mean, functionally.
In short, holding a non-voting preferred share with a stated dividend rate and a stated redemption value is mostly like holding a corporate bond [ie: a debt of the corporation that they must repay you], with a few key differences:
(1) In the case of bankruptcy, bonds have priority over any types of shares, and are repaid first - this makes them lower risk;
(2) Preferred shares likely have clauses that say that the corporation can forgo dividends in certain circumstances, whereas a bond's interest payment is guaranteed - this also makes bonds lower risk;
(3) Interest received on a bond likely has different tax consequences than dividends received from share ownership, both for the corporation and the investor [but what those differences are, will depend on jurisdiction];
(4) Preferred shares likely have no stated date of repayment, and often only repay when the corporation liquidates at the end of its life.
Because of the above, preferred shares often have higher dividend rates than corporate bonds have in interest rates, due to their different risk profiles.
Note that in all of the above, there is nothing special about owning '>50%' of a company's preferred shares - the person would simply have a larger investment, and no specific powers as a result.