The local ordinary of the outer space is the pope alone.
The following answer is written with respect to the laws of the Latin Church esp. the CIC. Most parts will apply also to Eastern Catholic Churches and the CCEO (just with a role for the Patriarchs).
That the pope has ordinary jurisdiction over the whole Church wherever it is, so also in outer space, is generally known.
can. 331 CIC
The bishop of the Roman Church, in whom continues the office given by the Lord uniquely to Peter, the first of the Apostles, and to be transmitted to his successors, is the head of the college of bishops, the Vicar of Christ, and the pastor of the universal Church on earth. By virtue of his office he possesses supreme, full, immediate, and universal ordinary power in the Church, which he is always able to exercise freely.
That there is no diocese for the outer space is also no suprise. I cannot prove absense, but you wont find such a diocese (and someone else would have found one before and answered the question). So the question comes down to: How is a Latin Catholic ruled at a place without a Latin Catholic hierarchy?
This question is not new, but very old. The high seas have no bishop and there was a time when nearly the whole world had no Catholic hierarchy. In the last century in most parts of the world at least a apostolic prefecture was errected. Maybe the last land place without a hierarchy is Antarctica (at least I never heared of a hierarchy for it). The only country without a Latin hierarchy is Eritrea. It only has the Eritrean Catholic Chruch.
There are two aspects of jurisdiction to look into: What particular law (local law) is applicable to someone living in area without jurisdiction? Who is the local Ordinary for the such a place?
In cann. 102-106 CIC the concept of a domicile is defined. This domicile is a dioecese and a parochy (or only a diocese). If one has no domicile or quasi-domicile in any diocese he is a transient (vagus, can. 100 CIC). So there is no bishop who could make particluar laws for him, but if he is (even for a short time) at a territory whithin a diocese he is bound to all laws of this place (can. 13 § 3 CIC). 
In some cases the local Ordinary of a place is needed. For example only the local Ordinary and pastor can ordinarily assist in a marraige (can. 1108 f. CIC). The term local Ordinary is defined in the CIC:
can. 134 §1. In addition to the Roman Pontiff, by the title of ordinary are understood in the law diocesan bishops and others who, even if only temporarily, are placed offer some particular church or a community equivalent to it according to the norm of can. 368 as well as those who possess general ordinary executive power in them, namely, vicars general and episcopal vicars; likewise, for their own members, major superiors of clerical religious institutes of pontifical right and of clerical societies of apostolic life of pontifical right who at least possess ordinary executive power.
§2. By the title of local ordinary are understood all those mentioned in §1 except the superiors of religious institutes and of societies of apostolic life.
As nobody else is the local Ordinary for a place without own hierarchy only the Pope is the local Ordinary for such a place. A marriage in space can only be held with the Pope in person or someone delegated by him. Note that the Roman Pontiff is local Ordinary and not the Apostolic See which would include the whole Roman Curia.
As stated before in the outer space is the same situation as in the high seas. For Pastoral Care for the People of the Sea the Pope made extra laws esp. the Motu Propio Stella Maris in 1997. The responsible authority in the Roman Curia is the Pontifical Council for the Pastoral Care of Migrants and Itinerant People. As the Pope did nothing like this for space missions there are no special rules and only the Pope has ordinary authority over the outer space.
The Pope makes also regularly rules for the Catholics who live in terretories without a hierarchy of their Church sui iuris, but with a hierarchiy of an other Catholic Church. As there is no Catholic hierarchy in space at all this does not help us anyway.
: Klaus Lüdicke (Editor): Münsterscher Kommentar zum Codex Iuris Canonici unter besonderer Berücksichtigung der Rechtslage in Deutschland, Österreich und der Schweiz. 53. Lieferung April 2017. Ludgerus Verlag. Essen, can. 107 Rn. 6 [most important German commentary on the CIC]
: ibid., can. 1109 Rn. 3