Creating a translation of a work protected by copyright without permission is an act of infringement, unless the creation of the translation falls within an exception to copyright. It does not matter whether this is done by a human translator, or by a computer translation program. A program is only a tool, and it is the act of translation, not the means by which it is accomplished that is an infringement.
There is not enough information in the question to determine if an exception to copyright, such as fair use in the US, or fair dealing in the UK, or news reporting or personal use in much of the EU, would apply. That is a highly fact-based determination, and also depends on the (unstated) country of jurisdiction. So for purposes of discussion I will assume that no exception applies, which seems likely from the limited information in the question.
Since the translation is not lawfully made, distributing it is probably a further act of infringement. This is true whether it is distributed to a limited but sizable audience, or to the general public. It is true whether the distribution is via streaming technology, or by posting on a web site, or both. It is true whether downloads are facilitated, or an attempt is made to prevent them. Any form of distribution is equally an infringement. This is true whether a fee is charged or not, although charging a fee might increase the damages that would be awarded in a suit.
the question reads:
The translation cannot be downloaded;
This does not matter, but it also cannot be fully accurate. The provider can refrain from providing an easy method of downloading the content. The provider can take steps designed to block downloads of the content. But if the content is so distributed that it can be read, it is possible for the receiver to capture and store it. Even if the content is displayed only as an image, with no associated text file, that image can be captured, put through OCR, and stored.
Another answer says:
If Google Translate is legal in the country or countries where Bob would operate the website, this should be equally not violative.
But Google Translate is a tool. It is the purpose to which it is put that makes its use infringing or not infringing. Using Google Translate, or any other translation program, to translate a work not protected by copyright is not infringing. Using a translation program in a way that falls under an exception to copyright, such as fair use, fair dealing, (in some countries) personal use, or another exception, is not infringement. Using a translation program to translate an entire book and then publishing the translation commercially would clearly be infringement. Many purposes for which people use routinely use Google Translate are probably infringing, just as many works posted on the internet are clearly infringing. Copyright owners do not choose to sue every possible infringer, the time and expense would be prohibitive. That does not make such uses non-infringing, and it does not make similar uses non-infringing. "But others do it all the time!" is not a defense.
The question is tagged "innocent-infringement" but I do not see how that could apply. An innocent infringer is a person who had no reason to believe that his or her act was an act of infringement. Here the website operator knows that the original content is protected by copyright. S/he knows that the use made probably does not fall within fair use or any other exception to copyright. S/he knows that the content is being provided to more than "a small group of people" or "a family circle". I do not see how this can be classed as innocent infringement.
The above conclusions should be valid under the law of any country that adheres to the Berne Copyright convention or the closely related TRIPS Agreement . Below I cite specific provision of US law that would apply. The laws of other countries on these points are mostly similar.
Specific Provisions from US Law (Title 17)
Derivative Work
Definition of “derivative work” from 17 USC 101:
A “derivative work” is a work based upon one or more preexisting works, such as a translation, musical arrangement, dramatization, fictionalization, motion picture version, sound recording, art reproduction, abridgment, condensation, or any other form in which a work may be recast, transformed, or adapted. A work consisting of editorial revisions, annotations, elaborations, or other modifications, which, as a whole, represent an original work of authorship, is a “derivative work”.
Exclusive rights
17 USC 106 provides in relevant part:
Subject to sections 107 through 122, the owner of copyright under this title has the exclusive rights to do and to authorize any of the following:
(1) to reproduce the copyrighted work in copies or phonorecords;
(2) to prepare derivative works based upon the copyrighted work;
(3) to distribute copies or phonorecords of the copyrighted work to the public by sale or other transfer of ownership, or by rental, lease, or lending;
Fair Use
17 USC 106 provides in relevant part:
Notwithstanding the provisions of sections 106 and 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright.
(I do not quote the statutory factors for analyzing fair use, because they are not specifically mentioned in the answer.)
Innocent Infringement
17 USC 504 (c) (2) provides in relevant part:
... In a case where the infringer sustains the burden of proving, and the court finds, that such infringer was not aware and had no reason to believe that his or her acts constituted an infringement of copyright, the court in its discretion may reduce the award of statutory damages to a sum of not less than $200. The court shall remit statutory damages in any case where an infringer believed and had reasonable grounds for believing that his or her use of the copyrighted work was a fair use under section 107, if the infringer was: (i) an employee or agent of a nonprofit educational institution, library, or archives acting within the scope of his or her employment who, or such institution, library, or archives itself, which infringed by reproducing the work in copies or phonorecords; ...
